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Linearization of bounded holomorphic mappings on Banach spaces. (English) Zbl 0747.46038
The author shows that for every open subset \(U\) of a Banach space there exists a unique Banach space \(G^ \infty(U)\) and a holomorphic mapping \(g_ u:U\to G^ \infty(U)\) such that every Banach valued bounded holomorphic function on \(U\) can be written as a composition of \(g_ u\) and a Banach valued continuous linear mapping on \(G^ \infty(U)\). This gives a linearization of bounded holomorphic mappings and shows that \(H^ \infty(U)\) has the structure of a dual Banach space. Applications to the study of holomorphic mappings of compact type, the approximation property and polynomials are given using this linearization result.
Reviewer: S.Dineen

MSC:
46G20 Infinite-dimensional holomorphy
46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
46B28 Spaces of operators; tensor products; approximation properties
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